Simple difference quotient question. Function -> simplify answer

[nukeman]

Simple difference quotient question. Function --> simplify answer

Homework Statement

I just don't understand this question! Here, I will take a picture of it. Can someone please explain how I answer these? Or even show me a website that shows how to do these? Its not in my review book.

QUESTION is in ATTACHMENTS!

Homework Equations

The Attempt at a Solution

 

[vela]

It seems pretty straightforward what the question is asking. What specifically don't you understand?

 

[nukeman]

Can you tell me the first step? When you look at this, what is the first thing you see/do?

 

[vela]

Exactly what the problem statement says: evaluate the difference quotient. If you don't understand what that means, try looking up "difference quotient" in your book.

 

[nukeman]

Must be missing something here.

I would plug in the f(x) values to the 2nd function, but there is no x's, just h ? How do I plug in values?

 

[Mark44]

The problem is asking you to evaluate the difference quotient to find f'(3).

 

[nukeman]

wow I'm lost. :(

 

[Macch]

Dude, f(x) = f(3+h) if x = 3+h

 

[Mark44]

wow I'm lost. :(

The difference quotient to find the derivative of a function, f'(x), looks like this:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

The difference quotient to find the derivative of a function at a specific number a, f'(a), looks like this:

$$ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

Do you recognize that your problem is asking for f'(3)?

 

[SammyS]

Homework Statement

I just don't understand this question! Here, I will take a picture of it. Can someone please explain how I answer these? Or even show me a website that shows how to do these? Its not in my review book.

Here's the image:

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So, if $f(x)=4+3x-x^2\,,$ then what is $f(3+h)\ ?$

 

[nukeman]

Here's the image:

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So, if $f(x)=4+3x-x^2\,,$ then what is $f(3+h)\ ?$

SammyS, that is the one part I am having trouble fully understanding...

 

[vela]

It would have helped if you could have said that right from the beginning.

In f(x)=4+3x-x², the x is what's called a dummy variable. Anything you put in the lefthand side where the x is shows up wherever the x appears on the righthand side. For example, suppose you have g(x) = 2x+4. Then
\begin{align*}
g(y) = 2y+4 \\
g(h+3) = 2(h+3)+4 \\
g(4z) = 2(4z)+4 \\
g(weasel) = 2(weasel)+4 \\
g(3) = 2(3)+4 = 10
\end{align*}
Now tell us what you think f(3+h) is equal to in your problem.

 

[Mark44]

SammyS, that is the one part I am having trouble fully understanding...

Apparently you don't understand how to evaluate a function's formula.

If g(t) = 3 - t² (for example)
then g(1) = 3 - (1)² = 3 - 1 = 2
and g(5) = 3 - (5)² = 3 - 25 = -23
and g(a + 2) = 3 - (a + 2)²

So for your function, what is f(3 + h)?

 

[nukeman]

"So for your function, what is f(3 + h)?"

f(3+h) = 4 + 3(3+h) - (3+h)^2 ?

 

[Mark44]

Yes. You'll need to expand the right side and collect common terms.

 

[vela]

Yup, that's it.